Kerry Back
BUSI 721, Fall 2022
JGSB, Rice University
Tesla went down 50% between Nov 2021 and May 2022.
It then went up 50% between May 2022 and Aug 2022.
Were Tesla shareholders back to even?
For each $100 of Tesla stock, shareholders experienced 100 → 50
and then 50 → 75.
They lost 25%, even though the average return was zero.
So, lose 50% and make 50% → lose 25%. Suppose you
make 50% and then lose 50%?
lose 50% and then make 100%?
make 100% and then lose 50%?
Given returns \(r_1, \cdots, r_n,\)
the geometric average return is the number \(r\) such that
\[(1+r)^{n}=(1+r_1)\cdots(1+r_{n})\]
So earning \(r\) each period produces the same accumulation as the actual returns \(r_1, \cdots, r_n.\)
We solve for it as
\[r=[(1+r_1)\cdots(1+r_n)]^{1/n}-1\]
The geometric average return is always less than the arithmetic average return.
\[\sqrt{1.5 \times 0.5}-1=-0.134\]
\[\sqrt{2 \times 0.5}-1=0\]